Generalized Boltzmann Equation

13 January 2010

The design of re-entry vehicles is subject to major challenges, which originate mainly from the three different flight regimes -- continuum, transition, and rarefied -- that govern the flight parameters. Whereas the Navier-Stokes equations provide a reasonable approximation to the physics of the continuum regime, other approaches are needed to model the remaining regimes. Kinetic approaches are widely used for the rarefied regime but are inapplicable for the other regimes. The Generalized Boltzmann equation however can cover all of the three regimes. As an integro-differential equation it can be solved by means of established numerics and it offers the flexibility to cover a wide range of characteristics of hypersonic flows such as polyatomic gases, particle interactions, and chemical reactions. Moreover, through the use of different potentials for inter-particle interactions (such as the Lennard-Jones potential), different species can be accurately simulated.

Compared with the Navier-Stokes equations the Boltzmann equation provides more information for points in the flow field. More precisely, not only the macroscopic state (density, bulk velocity, temperature) but a whole distribution function, defined on the space of all possible microscopic velocities, is given. From this function also non-equilibrium features such as heat flux and stress tensor can be derived.

A generic distribution function for a single point located in the flow field is exemplarily shown in the two figures.

Two dimensional Maxwell distribution which is the distribution function of a state in thermodynamical equilibrium; the center corresponds to the bulk velocity, the height is controlled by the density, and the breadth is influenced by the temperature.

"Broken'' Maxwell distribution function; without knowing the gradients of the macroscopic variables it can be specified to what extend this distribution deviates from a thermodynamical non-equilibrium state.